The conventional analysis of the coherence property due to the microbunching relies on paraxial wave equation [1]}, which cannot explain the quantum phenomenon of entangled photon pair emission. To overcome this problem, we develop a quantum collective double emission (QCDE) theory (see details in section 4 of SI). For simplicity, here we illustrate the physical picture of the coherent enhancement via an analogy with the quasi-phase-matching in nonlinear optics.
